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Saturday, January 30, 2010

The Meaning of Slope.

It seems to me that one of the most central ideas in a beginning Algebra I course is slope. When I first picked up the text book I was planning on using as the base for my 8th grade algebra course (College Preparatory Math), I noticed that slope was dealt with in several different locations during the year. With little experience actually teaching algebra, I found the idea of repeating a topic as "simple" as slope (rise over run) ridiculous.

Being of independent mind in a private school, I was free to reject, and so I did. I skipped the unit.

While I cannot directly correlate the results to the action, it did become evident, later that year, that my students' algebra education had taken a turn towards the "mechanistic". While many of them did successfully adopt the "rise over run" mantra, they began using funny language, such as identifying "x" as part of slope (think y=mx+b) or not getting the whole "change in y compared to the changed in x" concept.

I am in an unique position of being able to direct own mathematics program since I am also the 7th grade teacher. It wasn't until my second year into the job, though, that I realized the GRAND importance of "ratio" in a middle schooler's conceptual map of math. If they had a fluid notion of comparing two quantities, they were well positioned to simply transfer that idea to slope and move on. But, as is the case of many if not most of my students, they had a very creaky understanding of numerical comparisons and if they maintained a decidedly "fractional"view of "ratios" (part to whole), than indeed, the whole concept of "y" changing according to something done to "x" is confusing and meaningless.

As I study calculus with one of my super advanced 7th grader, I have finally come to see the supreme importance of slope in differenting equations. In fact, without an extremely solid foundation in the ratio known an "slope", calculus will remain, as it did for me so many years ago, a mystery wrapped in an enigma (but one in which I needed to get a passing grade).

This year's 8th graders are exhibiting even more "slope-itis" than in year's past (though this could be entirely due to my almost singular obsession with "figuring out what they know"). It was with this concern in mind that we embarked whole-heartedly into the once skipped unit on slope.

We hit a wall.

Indeed, lots of confusion. Many requests for clarification. A whole lot partial understandings or flat out misunderstandings.

It wasn't pretty.

I said to myself, "This can't go on. What to do?"

"Close up your books and let's go out on the green top"(thanks, private school, for the freedom to be really in the moment and not driven by outside schedules). All we took was a piece of paper and my iPhone (for the timer).

I asked for several volunteers to show us different style walking or running. Some sprinted, other ran backwards. Some strolled, others sashayed (and not only the girls, btw). In the end, we measured off 30 meters, chose 5 volunteers and had them, one by one, walk the distance. We timed them according to my iPhone stopwatch.

10 minutes later, we had a collection of data. We came into the room and created a table with the five volunteers' times. I asked "What is speed?" Many knew it was distance over time. I said, "So, that is a slope of line." Many of them looked confused. We made a big graph together and briefly reviewed that time would have to be the independent variable, always placed on the x axis, while distance would be the dependent variable, placed on the y axis.

From the table information we were able to create five different slopes. They graphed these lines and we talked about how steepness tells us something very important about "rate of change": steeper means faster change.

The class looked at me matter of factly. They claimed to know this already. In fact, judging from the discussion, they did know this. But I am convinced that they were not applying this sort of real world experience to the slope of lines we were analyzing in class.

On Friday afternoon I tweeted to my PLN: "it was a long week, but my students understand more about slope than when we started."